2024 JPJC Chapter 4 Functions

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 4 Function (Students’ version) / Pg 1 Chapter 4 : Functions Content Outline Include :  concepts of function, domain and range  use of notations such as 2f( ) 5x x  , 2f : 5x x  , 1f ( )x , fg( )x and 2f ( )x  finding inverse functions and composite functions  conditions for the existence of inverse functions and composite functions  domain restriction to obtain an inverse function  relationship between a function and its inverse Exclude the use of the relation 1 1 1(fg) g f   , and restriction of domain to obtain a composite function. Reference  http://www.h2maths.site  https://www.mathisfun.com/sets/function.html [Website on properties of function , composite functions]  Heinemann Higher Mathematics, Heinemann. Call No: CLA 510.712  MEI Pure Mathematics 2, 2nd ed, Hoddler & Stoughton. Call No: HAN 510  Modular Maths Pure Mathematics 2, Hoddler & Stoughton. Call No: SYK 510  DoBrilliantly ASMaths, Collins. Call No: GRA 510 Prerequisites  Sketch graphs y = f(x) of the following types: linear, quadratic, cubic, reciprocal, exponential, logarithmic, sine, cosine, and tangent.

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 4 Function (Students’ version) / Pg 2 Introduction Before we introduce functions, we need to understand what a relation is. In layman’s term, a relation is an equation connecting a variable x with another variable y.  A relation is a set of ordered pairs.  The first elements in the ordered pairs, the x-values, form the domain (Input values). Function is a relation between a set of inputs and a set of outputs with the property that each input maps to exactly one output. Relation Function Not a Function E.g. 3y x  x  1 1 2 y 2 4 5 E.g. 2y x x 0 4 9 y 0 2 or  2 3 or  3 1. Functions A function f : X Y is a rule (or relation) which associates each element in X (known as the domain) with a unique element in Y (known as the range), i.e. a function is a rule y = f(x) such that every value of x X has exactly one value of y Y . In diagrammatic form, a function can be visualised as follows:  The set of input values forms the domain.  The corresponding set of output values forms the range. Df is the notation for domain of f. Rf is the notation for range of f. f Df Rf X Y 1 1 2 2 4 5 X Y 9 0 4 3 3 0 2 2

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 4 Function (Students’ version) / Pg 3 1.1 To determine whether a given rule is a function A function f : X Y is a rule (or relation) which associates each element in X (the domain) with a unique element in Y (the range). This means that for the graph of a function, there will be only one y-value corresponding to every x-value. So a vertical line drawn at any x-value in the domain should cut the graph of the function at exactly one point. Hence, if there exists a vertical line ( x = k) which cuts the graph of the relation at more than one point (which means an x-value has more than one y-value), then the relation does not represent a function. Example 1 Determine which of the following represents the graph of a function and state your reason(s): (a) (b) If the GC can be used to plot the graph, use the command `p4 to plot the vertical line. Mini Exercise Write the following in inequalities and interval notations. Description of solution set Inequalities Interval notation values from 1 (inclusive) to infinity 1x values from negative infinity to 1 (exclusive) 1x values from 1 (inclusive) to 5 (exclusive) 1 5 x  all real values excluding 1 1 or 1x x  or \{ 1 } Note the use of interval notation :  ,a b to represent the interval  :x a x b   ,  ,a b to represent the interval  :x a x b   ,  ,a b to represent the interval  : ax x b   ,  ,a b to represent the interval  :x a x b   ,  ,a  to represent the interval  :x x a  , [ , )a  to represent the interval  :x x a  , ( , ) b to represent the interval  :x x b  , ( , ] b to represent the interval  :x x b  .  represents the set of real numbers.  represents the set of positive real numbers. 0  represents the set of positive real numbers including 0. [or {0} ] y x x y Solution: 1(a) There exists a vertical line, x k which cuts the graph at two points, hence it is not a function. 1(b) Any vertical line will cut the graph at most once. Hence this graph represents a function. O O

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 4 Function (Students’ version) / Pg 4 Example 2: The function f is defined by 2f: , x x 2x . State the rule and the domain of f. Solution: The rule is The domain of f = Note: There are various ways of writing a function as shown below: (a) p( x) = ex, x ≤ 0 (b) f : x  x2 , x  (c) 3g : , 0x x x  (d) q( x) = 1 , for 0 1 1 , for 1 0 x x x x         In most cases, the name of the function is denoted by lower case alphabet (f, g, p, q etc). All functions are defined by a rule and a domain. Example 3 Given f : x  (x + 1)2, x  . Determine whether g or h is the same as f. Give reasons for your answers. (a) g(u) = u2 + 2u + 1, u  (b) h(x) = (x + 1)2, x   Solution : Note: Two functions with the same rule but with different domains are considered as different functions.

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 4 Function (Students’ version) / Pg 5 1.2 Use of GC to find range The graph of a function is a way of visualising the relationship between y and x. The graph provides a simple way to see the range of a function. We can sketch the graph using the GC. Example 4 Find the range of the function 2g : 1, 2x x x   . Note to students: First , sketch the graph of y = x2 + 1, x > 2. [Refer to Annex 5.1] State the end point (2, 5). From the graph, determine the set of possible values of y that the function can take. Solution:  gR 5,  Example 5 Find the range of the function f: x  x2 +1, x 1 . Solution: Hence, Rf =  1, ( 1,2) 1 o (2, 5)

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023